701 
MECHANIC S. 
The centre of preffure is that point of a furface againft 
which any fluid preffes, to which if a force equal to the 
whole preffure were applied in a contrary direftion, it 
would keep the furface at reft. 
Prop. XI. To find the centre of preffure of a plane furface .— 
Let A B C D, fig. 9. be the furtace of the fluid, V W the 
plane, which being produced, let cd be its interfeftiou 
with the furface, P the centre of preffure, and G the 
centre of gravity ; and conceive the whole plane to be 
divided into an indefinite number of indefinitely-lmall 
parts, of which one is x ; draw PQ, Gg, xv, perpendi¬ 
cular to the furface, and P a, G m, xm, perpendicular to 
cd-, and join Qa, gn, vm\ then it is manifeft, that the 
triangles PQa, Ggn, xvm, are fimilar. Now the preffure 
on x perpendicular to V W is (Prop. IV.) asxxxv, and 
(Mechanics, Prop. XI.) its effeft to turn the plane about 
cd is as xyxvyxm-, but (fun. trian.) G n : Gg :: xm : 
G g 
x v — x m x -f- j hence, the effect of the preffure at x to 
G ft 
turn the plane about cd is as 
Gg 
Gn 1 
therefore 
G g 
G n 
Va — 
ftm of all the x x»x m 2 
the whole effeft is as the funt of all the xxxm 2 x 
But, if A = the area of VW, the preffure on VW is 
as AxGg-, therefbre the effect of that preffure at P, to 
turn the plane about c d, is as AxG^xPa- Hence, 
Gr 
Ax G gy'P a-=zfum of all thexyx na 2 X — i confequently 
Gs 
. Hence it appears, that P 
AX'J® 
is at the fame diftance from cd as the centre of percufllon 
is, cd being the axis of fufpenfion. They do not how¬ 
ever, in general, lie in the fame line, that is, in the line 
n G; for the efficacy of the preffure at x, to turn the plane 
about nG, is as xy.xvyt.mn, or (fince xv varies as xm) 
as *x xmy mn ; but the fum of all the xyxm ymn is not 
generally =0, therefore the whole preffure will not ne- 
ceffarily balance itfelf upon the line Gn. The fituation 
of the line aP muft therefore be determined, by making 
the fum of all the xXx rnXuin = o. 
The centres of preffure and percuffion do not therefore 
in general coincide, taking the centre of percuffion in its 
ufual acceptation. The centre of percuffion has always 
been defined to be that point in the line n G at which all 
the motion of the body would be deftroyed, eftimating 
the motion of the body about the line cd ; and the com¬ 
putations have been always made upon this principle. 
But the body, after its aftiou againft that centre, may 
ftill have a tendency to turn about the line n G. If there¬ 
fore we were to define the centre of percuffion to be that 
point at which the whole motion of the body would be 
deftroyed, the centres of preffure and percuffion would 
riot, in general, coincide ; in which cafe, the pofition of 
the line a P muft be computed on the above principle. 
Of SPECIFIC GRAVITY. 
That quality of bodies which is defignated by the term 
fpecific gravity, has already been defined in p. 701. The 
object of the prefent feition is to explain the principles 
on which the different methods of afeertaining the fpe¬ 
cific gravities of folids and fluids are founded, and to 
give fome account of the beft of thofe methods. Previous 
to which it may be proper to make a few obfervations, 
naturally deduced from the definition of fpecific gravity 
and the nature of bodies in general. 
r. The fpecific gravities of bodies are in the fame pro¬ 
portion as their weights, when the magnitudes of the bo- 
d es are equal. 
2. Where the weights of the bodies are equal, the fpe¬ 
cific gravities are inverfely as their magnitudes. 
3. When the fpecific gravities are equal, their weights 
are directly as their magnitudes. 
4. When neither the magnitudes nor the fpecific gra¬ 
vities are equal, the weights of bodies are as their magni¬ 
tudes and fpecific gravities conjointly. 
W 
To exprefs thefe relations algebraically, let — repre- 
w W 
fent the ratio of the weights of two fubftances, — the 
S m 
ratio of their magnitudes, and — that of their fpecific 
gravities ; then will the general relation of thefe quanti- 
• . ^ , , . , . W M S 
ties be expreflea by the equation —=—X —. In eftimat- 
w m s 
ing the weights, magnitudes, and fpecific gravities, of 
fubftances, fome ftaudard-quantities are always aflumed 
to which other bodies are referred : the letters w, m, and s, 
may reprefent thefe ftandards, and each of them might 
be aflumed equal to 1 ; but fuch affumption would not 
correfpond with the meafures and weights now in life. It 
will, therefore, be more eligible that m fnould reprefent 
the magnitude of fome known meafure which may be 
affumed for unity, as a cubic inch, a cubic foot, &c. and 
s any convenient number in the geometrical progreffion 
1, 10, 100, 1000, See. Now it has been found, that the 
denlity of rain-water is more nearly uniform, in different 
circurnttances of time and place, than any other fubftance, 
whether-lblid or fluid; and by a fortunate coincidence it 
happens, that the weight of a cubic foot of rain-water 
is exactly 1000 ounces avoirdupoife. If, then, we make 
w=z 1000, m~t, and s=iooo, we (hall obtain WziMxS, 
that is, the weight of any body in avoirdupoife ounces 
will be equal to the product of the magnitude in cubic 
feet into the fpecific gravity taken from that fcale in 
which the fpecific gravity of rain-water is 1000. Hence 
it appears, that a knowledge of the fpecific gravities of 
homogeneous bodies will enable us to determine their 
weight, without aftually weighing them, provided we can 
afeertain their magnitudes : and converfely, however irre¬ 
gular the (hape of bodies may be, if we know their weights 
and fpecific gravities we may readily determine their mag¬ 
nitudes in feet, namely, by dividing the weight in avoir¬ 
dupoife ounces by the fpecific gravity, or by the weight 
of a cubic foot in avoirdupoife ounces. 
But in philofophical fubjeffs, the weights of bodies, 
being for the molt part fmall, are eftimated in troy ounces 
or grains, the magnitudes being referred to a cubic inch 
as a ltandard. Now a troy ounce is to an avoirdupoife 
. . , 1000 4.375 
ounce as 480 to 437^5 confequently--x —7— =‘52740 
1728 400 
of an ounce troy, or 253-181 grains, the weight of a cubic 
inch of water. And hence the magnitude of a folid* 
VV 
eftimated in cubic inches, is =-—— > when the weight 
W 253-181 S 
W is in grains; or=-—, when the weight is known 
b -52746 S 
in troy ounces. And converfely, the weight eftimated in 
grains == 253-181 xMxS, when the magnitude M is in 
cubic inches; and, if eftimated in troy ounces, Wss 
•52746XMXS. In all thefe cafes S being the fpecific gra¬ 
vity, in terms correfponding to 1000, for that of rain¬ 
water. 
Hence alfo we may (how how to determine very accu¬ 
rately the diameter D of any fmall fphere, whofe fpecific 
gravity is S (to that of water 1000), its weight W being 
known in grains. For the content of a fphere whofe dia¬ 
meter is 1 being *523598, we have 1 : 0-523598 :: 253*181 
gr. (the weight of a cubic inch of water) : 132-5648 gr. 
the weight of a fphere of water whofe diameter is 1 inch. 
Therefore, fince the weights are as the magnitudes and fpe¬ 
cific gravities conjointly, and the magnitudes of fplieres 
are as the cubes of their diameters, we have 132*5648 . 
S /W 
D z .-= W; whence we find D = 1-961208 3 aI—. 
1000 ^ S 
After a manner not very widely different, may various 
other ufeful rules and theorems, applicable to the admea- 
furement of bodies either regular or irregular, be eafily 
deduced. 
